Piccard group

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It is different from "" [[Piccard modular group (English version)]].

In the mathematics, I write the Piccard group of space X with the ring (Picard group) as Pic(X) and am the group which the same model of the reversible layer (or a straight line bunch) on X having the group action that is the tensor product does. It is a global version of the constitution of factors group and the ideals group, and this constitution is used by algebra geometry and the theory of double bare manifolds well.

Piccard group is layer cohomology group

${\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})}$

I can define としても.

For 整 scheme (integral scheme), the Piccard group can show that it is kind group and the same model of the Cartier factor. For double bare manifolds, the series of index layer gives basic information of the Piccard group.

From a theory of Emile Picard (Émile Picard) particularly the theory of the factor of the algebra curve, the name of Piccard is on.

Table of contents

Example

• Piccard group of the spectrum of the Dede quinte domain is ideals group of the Dede quinte domain.

• The reversible layer of projection space Pⁿ(k) of body k is a layer a torsion (English version) (Twisting sheaf) ${\displaystyle {\mathcal {O}}(m),\,}$  であるから, the Piccard group of Pⁿ(k) are the same model in Z.

• The Piccard group of the アフィン straight line to have two origins on k toward is the same model in Z.

Piccard scheme

The Piccard scheme (Picard scheme) that is constitution of the Piccard group (can express it in the version of box hand (English version) (representable functor)) scheme structure is the step that is important in 双対理論 (English version) (duality theory of abelian varieties) of algebra geometry particularly the Abel manifold. The Piccard scheme is constructed in Grothendieck (1961/62), and there is a mention in Mumford (1966) and Kleiman (2005) again. Piccard manifold is 双対 of the アルバネーゼ manifold of the classic algebra geometry.

When it is the most important by classic algebra geometry, it is written the non-specific perfection manifold (English version) on 0 bodies to the connection ingredient of the identity element of the Piccard scheme (complete variety) with Pic⁰(V) for V, and 標数 is Abel manifold. A shop of V flatters it, and particularly this ingredient that V is a curve is a manifold. However, in 正標数 Junichi Igusa (English version) a cover about having Pic⁰(S); therefore constituted an example of smooth projection curved surface S which did not become the Abel manifold.

Quotient ${\displaystyle \operatorname {Pic} (V)/\operatorname {Pic} ^{0}(V)}$  は [limited generation Abelian group]] I am called ネロン セヴィリ group of であり, V and write it as NS(V). In other words, the Piccard group adapts to the next perfect sequence.

${\displaystyle 1\to \mathrm {Pic} ^{0}(V)\to \mathrm {Pic} (V)\to \mathrm {NS} (V)\to 0.\,}$ 

The fact that a rank is limited is a base theorem of Francis セヴィリ (English version) (Francesco Severi) (theorem of the base). The rank is written as ρ(V) where it is often it which is the number of Piccard of V (Picard number). In NS(V), the algebraic equivalent (English version) of the factor on V describes a kind (algebraic equivalence) geometrically. In other words, it is discrete invariant and is easy to treat the classification when I use strong non-linear equivalence relation by a substitute for primary system (English version) (linear equivalence of divisors) of the factor. The algebraic equivalent is related closely with numerical equivalent (English version) (numerical equivalence) which is the classification that is topology Cal essentially by the number of the crossing-over.

Relative Piccard scheme

f: X → S of the scheme shoot; とする. For S-scheme T where is arbitrary as for relative Piccard box hand (relative Picard functor) (if or it is a scheme relative Piccard scheme),

${\displaystyle \operatorname {Pic} _{X/S}(T)=\operatorname {Pic} (X_{T})/f_{T}^{*}(\operatorname {Pic} (T))}$ 

I am given により [¹]. Here,${\displaystyle f_{T}:X_{T}\to T}$  It is a base change of は f, and fT ^* is the pull return.

(when a degree is defined for Piccard group X_s, for geometric generation point s → T of ,) all, I return a pull of L along s ${\displaystyle s^{*}L}$  But, if, as a reversible layer on fiber X_s, it is degree r,${\displaystyle \operatorname {Pic} _{X/S}(T)}$  I say that の L is degree r.

Reference item

• Layer cohomology
• Cartier factor
• Masanori line bundle
• Ideals group
• アラケロフ group (English version) (Arakelov class group)

References

• Grothendieck, A. (1961/62), V. Les schémas de Picard. Théorèmes d'existence, Séminaire Bourbaki, t. 14,, http://www.numdam.org/item?id=SB_1961-1962__7__143_0 
• Grothendieck, A. (1961/62), VI. Les schémas de Picard. Propriétés générales, Séminaire Bourbaki, t. 14,, http://www.numdam.org/item?id=SB_1961-1962__7__221_0 
• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052 
• Igusa, Jun-Ichi (1955), "On some problems in abstract algebraic geometry," it is Proc. Nat. Acad. Sci. U.S.A. 41: 964–967, doi: 10.1073/pnas.41.11.964 
• Kleiman, Steven L. (2005), "The Picard scheme," it is Fundamental algebraic geometry, Math. Surveys Monogr., 123, Providence, R.I.: American Mathematical Society, pp. 235–321, arXiv:math/0504020, MR 2223410 
• Mumford, David (1966), Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, 59, Princeton University Press, ISBN 978-0-691-07993-6, MR 0209285, OCLC 171541070 
• Mumford, David (1970), Abelian varieties, Oxford: Oxford University Press, ISBN 978-0-19-560528-0, OCLC 138290 

Footnote

1. ^ Kleiman 2005, Definition 9.2.2.

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